Master Thesis - Department of Computer Science

4.2 Obtaining Eigenmodels in Range Space and Null Space of Within-
class Scatter
In appearance based face recognition techniques, a face image of w × h pixels is
represented by a vector in a d (= wh)-dimensional space. Therefore, each face image
corresponds to a point in d-dimensional image space. Let the training set be defined as
X = [x1 1 , x12 , ..., x1N , x21 , ..., xCN ], where C is the number of classes and N is the number
of samples per class. An element x i j denotes jth sample from class i and a vector
in d-dimensional space R d . Total number of training samples is M = NC. Then
within-class (Sw), between-class (Sb) and total (St) scatter matrices can be defined
as,
Sw =
Sb =
St =
C� N�
(x
i=1 j=1
i j − µi)(x i j − µi) T , (4.1)
C�
N(µi − µ)(µi − µ)
i=1
T , (4.2)
C� N�
(x
i=1 j=1
i j − µ)(x i j − µ) T = Sw + Sb. (4.3)
where µ is the overall mean and µi is the mean of i th class. Methods searching for
discriminative directions only in null space of Sw try to maximize a criterion given
as,
J(W Null
opt ) = arg max
|W T |W
SwW |=0
T SbW |
= arg max
|W T |W
SwW |=0
T StW |. (4.4)
Similarly, the directions providing discrimination in range space can be searched by
maximizing a criterion J(W Range
opt ) which is defined as,
J(W Range
opt ) = arg max
|W T |W
SwW |�=0
T SbW |. (4.5)
To find the optimal projection vectors in null space, we project all samples in null
space and then obtain W Null
opt
by applying PCA. As the basis vectors constituting null
space do not contain intra-class variations, we obtain single common vector for each
class. Any sample as well as mean of a class will provide the same common vector for
that class after projection in null space. The same does not happen for range space
as it contains the complete intra-class variations present in the face space. So, we
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